8  Completing Reaction Engineering Tasks

To complete common reaction engineering tasks, reactor models must be generated and the equations in them must be solved. Chapter 6 showed how to generate the reactor design equations for the four ideal reactor types, and Chapter 7 described four types of reaction engineering tasks. This chapter presents a general workflow that is used throughout the remainder of the book to complete reaction engineering assignments.

8.1 Summarizing Reaction Engineering Assignments

In a book like this, reaction engineering assignments take the form of a brief narrative that provides data and then assigns tasks and/or asks questions. An assignment summary provides a concise summary of the information in the assignment narrative. A good summary makes it possible to complete the assignment without needing to refer back to the assignment narrative.

The information provided in an assignment narrative will vary depending upon the specific tasks being assigned. As such, the items listed here will not all appear in any one assignment summary. The order of the items in the summary is not important.

Reaction(s)

Every reaction occurring in the system being analyzed should be listed and numbered.

Rate Expression(s)

If known or proposed, the rate expression for each reaction above should be listed and numbered.

Reactor System

Each reactor present in the system being analyzed should be listed. For each one, its type and operational details should be included.

Reactor System Schematic

A schematic diagram of the system being analyzed should be included. Each flow stream and piece of equipment should be given a unique label that can be used as a subscript on variables appearing in the model equations.

Adjusted Experimental Inputs

When experimental data are being analyzed, quantities that were varied from one experiment to the next should be identified.

Stimulus Type

If stimulus-response data are being analyzed, the type of stimulus should be listed.

Experimental Response

When experimental data are being analyzed, quantities that were measured in the experiments should be identified.

Quantities of Interest

Any variables for which the assignment requests values should be listed. If graphs or tables are requested, they should be listed along with the quantities needed to generate them. If an assessment is requested, the data needed to make it should be listed.

Parameters

If the assignment entails varying parameters or estimating the values of parameters, the parameters should be listed. If values of parameters to be varied are specified, they should be included in the listing.

Given and Known Constants

Each quantity for which the assignment narrative provides a value should be listed, along with the given value.

Basis

If all of the values provided in the assignment narrative are intensive, a value can be chosen for one extensive variable. The chosen value is called the basis, and it should be included in the assignment summary.

8.2 Generating Model Equations

The equations for modeling a reactor or other piece of equipment can be arbitrarily classified as model equations, supporting equations, and ancillary equations. All three are considered in this section.

8.2.1 Model Equations

Generally, a set of model equations should be generated for each reactor in the system. If additional equipment is present in the system, and if the mole and energy balances for that equipment were not incorporated in a reactor model, a set of model equations should be generated for each piece of equipment, too. In simpler systems, there will only be one set of model equations, and they will be the design equations for a reactor. The remainder of this section is written in terms of a simple system like that. Systems with more than one set of model equations will be considered in later chapters.

The assignment summary should include all of the information needed for generating the model equations for the reactor. Chapter 6 describes how to generate the reactor design equations for each reactor in the system starting from the appropriate mole, energy and momentum balances. The model equations should be simplified as appropriate for the system being analyzed. The model equations will be a set of ATEs, IVODEs, DAEs, or BVODEs. The type of equations in the set should be identified (see Appendix F) along with the variables to be found by solving them.

8.2.2 Supporting Equations

Supporting equations are necessary for the numerical solution of each of the four types of model equations. If the model equations are ATEs, supporting equations defining an initial guess for the ATE variables should be added to them. If the model equations are IVODEs, supporting equations defining the initial and final values should be added to them. If the model equations are DAEs (coupled sets of ATEs and IVODEs), supporting equations defining both an intial guess for the ATE variables and initial and final values for the IVODE variables should be added to them. Finally, if the model equations are BVODEs, supporting equations defining the initial mesh and the initial guess for the dependent variables should be added to them.

8.2.3 Ancillary Equations

Additional unknowns appearing in the model and supporting equations also should be identified. Additional unknowns are quantities other than the variables to be found by solving the equations, known constants, and parameters. When solving the model equations numerically, the values of the additional unknowns must be calculated within computer functions called residuals functions and derivatives functions (see Appendix F). More specifically, each additional unknown must be calculated using the model equation variables and known constants. Ancillary equations for calculating all of the additional unknowns therefore should be added to the model equations.

8.3 Calculating the Quantities of Interest

The model equations described above can be solved to find the values of the reactor variables. The reactor variables are things like the reacting fluid volume, outlet or final molar amounts, outlet temperatures, etc. Often these are not the quantities of interest for the assignment. In that situation, the reactor variables must be used to calculate the quantities of interest. Therefore, in addition to the reactor model equations, the equations for calculating the quantities of interest should be summarized separately. The variables found by solving the reactor model equations can be used in these equations.

8.4 Formulating Numerical Calculations

In some scenarios the reactor design equations can be solved analytically, but in many more situations, analytical solution is not possible. Numerical solution is necessary for many important, real-world situations that involve adiabatic operation, multiple reactions, complex rate expressions, non-isothermal operation involving heat exchange, and others. Therefore, in Reaction Engineering Basics the reactor design equations are always solved numerically, even if analytical solution is possible.

Different readers of Reaction Engineering Basics may choose to use different mathematics software when completing assignments. For this reason, code for performing numerical calculations is not presented nor provided in the book. Fortunately, computer functions for solving sets of a given type of equations all require essentially the same input, irrespective of the specific mathematics software package they come from. This makes it possible to formulate the calculations needed to complete an assignment by writing specifications for the functions that are needed. Then the specifications can be used to write the actual code using whatever software package one chooses.

The assignment summary, the models generated as described above, and the summary of equations for calculating the quantities of interest provide all of the information needed for formulating the numerical calculations. When formulating the calculations it is assumed that known constants are available whenever they are needed. The formulation then consists of specifications for three or four computer functions, depending on the type of equations being solved.

8.4.1 Specifications for Residuals and Derivatives Functions

As described in Appendix F, residuals expressions effectively provide a set of ATEs to a mathematics software package that will solve them and derivatives functions similarly provide a set of ODEs. If the reactor model equations are ATEs, the formulation of the calculations must include specifications for a residuals function. If the reactor model equations are IVODEs, the formulation of the calculations must include specifications for a derivatives function. If the reactor model equations are DAEs or BVODEs, the formulation of the calculations must include specifications for both a residuals function and a derivatives function.

8.4.1.1 Specifications for a Residuals Function

The first item in the specifications for a residuals function lists the arguments that are passed to it. Because the residuals function is called by the mathematics software package, its only argument is always a guess for the values of the ATE variables.

If the residuals function requires other input variables for calculating its return values (below), the second item in the specifications lists the other input variables as “must be available.” This serves as an alert when actually writing the computer code that these variables must be made available to the residuals function by some means other than passing them as an argument.

The next item in the specifications for a residuals function lists the variables it returns to the function that called it. Again because the residuals function is called by the mathematics software package, it returns only the ATE residuals corresponding to the guess for the ATE variables that was passed to it as an argument. Residuals are defined in Appendix F.4.1. When the residuals equal zero, the guess used to calculate them is a solution of the ATEs.

The final item in the specifications for a residuals function is its algorithm. Here “algorithm” is taken to mean the sequence of equations used to calculate the return values, not a computer logic diagram. Starting with the guessed values of the ATE variables and the variables that have been made available to it, the algorithm first uses the ancillary equations to calculate each of the additional unknowns in the ATE model equations, and then it calculates the residuals.

8.4.1.2 Specifications for a Derivatives Function

A derivatives function is analogous to a residuals function, and its specifications consist of the same items. The only arguments are always a value for the independent variable and a value for each of the dependent variables. If the derivatives function requires other input variables for calculating its return values (below), they are listed as “must be available.”

A derivatives function returns only the derivatives of the dependent variables corresponding to the independent and dependent variables that were passed to it as arguments. The evaluation of the derivatives of the dependent variables is discussed in Appendix F.4.2.

The algorithm for a derivatives function starts with the values of the independent and dependent variables passed to it and the variables made available to it. It first uses the ancillary equations to calculate any addition unknowns in the IVODE model equations, and then it evaluates the derivatives.

8.4.2 Specifications for Reactor Functions

Reactor functions solve the reactor model equations to find and return the reactor model variables. Reactor functions are not called by the mathematics software package, so their specification do not include a list of variables that “must be available.” The arguments passed to a reactor function are selected by the engineer who is solving the reactor model equations. If the reactor model equations are going to be solved multiple times using different values of one or more parameters, a value for each parameter is often passed to the reactor function.

In Reaction Engineering Basics, reactors almost always return the reactor model variables found by solving the reactor model equations. The algorithm for a reactor function uses the supporting equations to define arguments that must be provided to the solver (a function from a mathematics software package function that will solve the model equations). It makes necessary variables available to the residuals or derivatives functions, calls the solver, and returns the solution found by the solver.

8.4.3 Specifications for Deliverables Functions

Specifications for one more function must be included in the formulation of the calculations. This function uses the results from solving the reactor model equations to calculate the quantities of interest. In Reaction Engineering Basics it is called the deliverables function.

Typically it takes no arguments and doesn’t return anything. Its algoritm calls the reactor model function to get the reactor model variables and calculates the quantities of interest using the equations listed earlier. If the assignment involves an optimization task or repeating calculations while varying one or more parameters, the deliverables function will vary the parameters and call the reactor function as necessary. It may also display results on screen and save results to files.

8.5 Implementing and Executing Calculations

The assignment summary and the formulation of the calculations will include all of the information that is needed to write the code to perform the calculations using the mathematics software package of one’s choice. In Reaction Engineering Basics, this stage of the workflow is mentioned, but code is not shown for previously noted reasons.

8.6 Presenting and Discussing Results

The final component of the assignment completion workflow described here involves discussing the results of the calculation and presenting the quantities of interest as requested in the assignment narrative. While it is often tempting to simply report the results, a better approach includes discussion of the results.

For some calculations, an initial guess is provided to the solvers that solve the model equations. If the specified guess did not yield a converged solution, a discussion of how the guess was modified and the reasoning behind it can be useful. Similarly, if the assignment requested an assessment of the accuracy of a model that was fit to experimental data, the reasoning behind the final assessment can be included.

It is also worthwhile to discuss whether the results appear to be reasonable. This is particulary relevant when one is just learning reaction engineering. It causes the engineer to think about the results, and helps develop a greater depth of understanding. Discussions of this kind are often supported by a qualitative analysis of the system as described in Section 7.1. Every assignment is different, but there is almost always some aspect of the results or conclusions that merits further discussion.